Motion perpendicular to the direction of wave velocity: time derivatives of the wave function.
The motion of a single point on a vibrating string is purely vertical for a transverse wave traveling horizontally.
A point on a vibrating string moves in simple harmonic motion perpendicular to the direction of the wave velocity.
Given the wave function for the transverse wave, y(x,t)=Acos(kx-wt), we can find the velocity and acceleration using time derivatives of the wave function.
The velocity is given by the first partial derivative with respect to time giving us a velocity function of Awsin(kx-wt).
The acceleration is given by the second partial derivative with respect to time, giving us an acceleration function of -Aw^2cos(kx-wt).
